Mean-value theorems of multiplicative functions on additive arithmetic semigroups
By an additive arithmetic semigroup it is understood a free commutative semigroup \(G\) with identity element \(1\) such that \(G\) has a countable free generating set \(P\) of ``primes'' \(p\) and such that \(G\) admits an integer-valued degree mapping \(\partial\) which satisfies some suitable conditions. Let \( G(n)\) denote the total number of elements of degree \(n\) in \(G\). In the paper under review the author assumes that \(G(n)\) as \(n\) tends to infinity behaves like \[ G(n)= q^n\sum_{\nu=1}^rA_\nu n^{\rho_\nu-1}+O(q^{n}n^{-\gamma}). \] with some restrictions on \(q,A_\nu,\rho_\nu,\gamma\). He deduces analogs of the well-known Erdős-Wintner theorem and the Halász-Wirsing-Delange theorem, but the exact formulations of his results are too complicate to be given here explicitly. The key result is Theorem 1.1 in which the author considers the asymptotic behavior of the generating function \(\widetilde F(y)\) as \(| y| \to q^{-1}\) depending on that of \(F(m)\) where \(F\) is the summatory function of a multiplicative function \(f\) defined on \(G\). As is well-known, in the classical theory, the values \(f(2^k)\) require special considerations. The same is true in this more general situation, but here the constraints are \(\partial(p)\leq \frac{\log 2}{\log q}\). [Editor's remark: The corrigendum concerns the false theorems 6.4.2 and 1.6.]
scientific article; zbMATH DE number 1268067